(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

(* The type of positivstellensatz -- used to communicate with sos *)
open Num

type vname = string;;

type term =
| Zero
| Const of Num.num
| Var of vname
| Inv of term
| Opp of term
| Add of (term * term)
| Sub of (term * term)
| Mul of (term * term)
| Div of (term * term)
| Pow of (term * int);;


let rec output_term o t =
  match t with
    | Zero -> output_string o "0"
    | Const n -> output_string o (string_of_num n)
    | Var n   -> Printf.fprintf o "v%s" n
    | Inv t   -> Printf.fprintf o "1/(%a)" output_term t
    | Opp t    -> Printf.fprintf o "- (%a)" output_term t
    | Add(t1,t2) -> Printf.fprintf o "(%a)+(%a)" output_term t1 output_term t2
    | Sub(t1,t2) -> Printf.fprintf o "(%a)-(%a)" output_term t1 output_term t2
    | Mul(t1,t2) -> Printf.fprintf o "(%a)*(%a)" output_term t1 output_term t2
    | Div(t1,t2) -> Printf.fprintf o "(%a)/(%a)" output_term t1 output_term t2
    | Pow(t1,i) -> Printf.fprintf o "(%a)^(%i)" output_term t1 i
(* ------------------------------------------------------------------------- *)
(* Data structure for Positivstellensatz refutations.                        *)
(* ------------------------------------------------------------------------- *)

type positivstellensatz =
   Axiom_eq of int
 | Axiom_le of int
 | Axiom_lt of int
 | Rational_eq of num
 | Rational_le of num
 | Rational_lt of num
 | Square of term
 | Monoid of int list
 | Eqmul of term * positivstellensatz
 | Sum of positivstellensatz * positivstellensatz
 | Product of positivstellensatz * positivstellensatz;;


let rec output_psatz o = function
  | Axiom_eq i -> Printf.fprintf o "Aeq(%i)" i
  | Axiom_le i -> Printf.fprintf o "Ale(%i)" i
  | Axiom_lt i -> Printf.fprintf o "Alt(%i)" i
  | Rational_eq  n -> Printf.fprintf o "eq(%s)" (string_of_num n)
  | Rational_le  n -> Printf.fprintf o "le(%s)" (string_of_num n)
  | Rational_lt n  ->  Printf.fprintf o "lt(%s)" (string_of_num n)
  | Square t       -> Printf.fprintf o "(%a)^2" output_term t
  | Monoid l       -> Printf.fprintf o "monoid"
  | Eqmul (t,ps)   -> Printf.fprintf o "%a * %a" output_term t output_psatz ps
  | Sum (t1,t2)    -> Printf.fprintf o "%a + %a" output_psatz t1 output_psatz t2
  | Product (t1,t2)    -> Printf.fprintf o "%a * %a" output_psatz t1 output_psatz t2